Method for managing risk in markets related to commodities delivered over a network

ABSTRACT

A system, method, software, and portfolios for managing risk in markets relating to a commodity delivered over a network are described, in which a market participant constructs portfolios of preferably liquid price risk instruments in proportions that eliminate the Spatial Price Risk for the market participant&#39;s underlying position. Techniques are also disclosed for constructing and evaluating new price risk instruments and other sets of positions, as well as identifying arbitrage opportunities in those markets.

RELATED APPLICATION

The present application claims the benefit of U.S. Provisional PatentApplication Ser. No. 60/123,823 entitled, “A New Business Procedure forElectricity Markets: Hedging Against Transmission Congestion Risk inLocational Spot Markets, Detecting Arbitrage Opportunities, Valuing andConstructing Synthetic Locational Futures Contracts and Valuing andConstructing Locational Options and Other Derivatives,” filed on Mar.11, 1999 by Fernando L. Alvarado and Rajesh Rajaraman, the contents ofwhich are incorporated by reference in their entirety.

FIELD OF THE INVENTION

The present invention relates to financial services and moreparticularly to a system and method for managing risk in markets relatedto commodity delivery over a network.

BACKGROUND OF THE INVENTION

For many years, electric power and communications utilities operated ina highly regulated market. As these and similar industries arerestructured, deregulated, and created, new competitive commoditymarkets are coming into existence in which prices are determined bysupply and demand. For example, regulated utilities historically soldwholesale power under cost-based tariffs with retail prices set on acost-plus fixed-return basis. As a result, these utilities had noincentive to manage the risk of potential changes in the price of thepower they generated. Users of power similarly had no incentive tomanage price risk because they had no control over the price they paidfor power. The shift to a competitive market, however, has created anincreasing awareness of electricity price risk and the need for managingthe price risk.

The need for managing the price risk of electricity is greater than inmany other markets because there is a high variation in the price ofelectricity over both time and space. There is a high variation in theprice of electricity over time because it is difficult to store electricpower, necessitating that the electricity be produced when demanded.Even under normal conditions, electricity prices may fluctuate widelyover the course of a day.

The high variation in the price of electricity over space is due to thephysical nature of the power network. The power flow over a particulartransmission line between two locations in an electric power networkcannot be directly controlled due to the laws of physics, according towhich electric power flows over all possible paths in accordance withtheir impedance. For example, the 1989 Federal Energy RegulatoryCommission (FERC) transmission task force discovered that as much as 50percent of a power transfer from Ontario Hydro to the New York PowerPool may have used transmission lines that were hundreds of miles awayfrom the direct interconnection between the two locations. As a result,when electric power is transferred into or out of the power transmissiongrid, that transfer of power may affect the distribution of electricityon any transmission line in the network.

Congestion in the transmission system can have a significant effect onthe price of electricity. When one transmission line in the network isloaded to its full capacity, power cannot be rerouted over a differenttransmission line to avoid the congested line. Even if the transmissionof power is congested between only two locations, that congestionaffects the prices of electricity at other locations in the network. Theprice of electricity downstream of the congested line tends to increase,encouraging additional power generation to be brought on line to servethe load downstream of the congested line. Meanwhile, the price ofelectricity upstream of the congested line will tend to decrease,discouraging power generation upstream of the congested line.

Various approaches have been proposed to manage the price risk ofelectricity. For example, a generator can hedge against the risk thatthe price of electricity will fall at a particular location electricityvia a forward contract. A power forward contract is a privatelynegotiated agreement between commercial parties containing a bindingobligation to deliver electricity at a specified location and price. Asignificant disadvantage of forward contracts is that the market forforward contracts can be illiquid at particular locations. Forwardmarkets achieve higher liquidity by concentrating the market activityinto a few standard locations. There are thousands of differentlocations in the power network but only a few locations in which anyforward liquidity exists. Therefore, it may be difficult for thegenerator to find a willing buyer of the forward contract at anacceptable price at their specific location.

Futures contracts are generally standardized contracts for the deliveryof a commodity (here, electricity) in the future at a price agreed uponwhen the contract is made. Because futures contracts are used primarilyfor hedging against price risk or speculating on the price of thecommodity, market participants typically close out their futurescontracts positions financially rather than through delivery. In the PJM(Pennsylvania, New Jersey, and Maryland) market, which has over 1000locations, an electricity futures market currently exists for deliveryonly at the location PJM West.

Because the location for which a liquid forward and futures marketexists is typically not the same location at which a particular marketparticipant, such as a power generator, would like to make or takedelivery, market participants using forward contracts to hedge theirunderlying positions incur basis risk because prices at differentlocations are not consistent. This basis risk is sometimes referred toas “Spatial Price Risk.” For example, due to congestion, the price ofelectricity at one location may differ from the price of electricity atthe liquidly traded location.

Besides forward and futures contracts, other price risk managementcontracts include price swaps, basis swaps, option contracts, andcongestion compensation contracts. The first three types of riskmanagement contracts are well-known outside of the wholesale electricitymarket. A congestion compensation contract explicitly compensates one ofthe parties if there is congestion on a transmission line. Various kindsof congestion compensation contracts have been proposed and are knownunder various names.

For example, a Transmission Congestion Contract (TCC) is a congestioncompensation contract for buying power at one location and deliveringthe same amount of power at a different location at a specified price.The TCC pays if there is a difference in price between the twolocations, or, in other words, if there is a congested line in the powernetwork. A TCC, however, suffers from a lack of liquidity because thereare thousands of locations in the power network, but relatively fewmarket participants interested in a particular location.

As another example, Stoft proposed a futures contract, not on the priceof electricity at a particular location, but on an explicit congestionprice for delivering electricity between two locations. The explicitcongestion price values the use of scarce transmission resources, suchas a congestible transmission line. A disadvantage of this approach isthat the market for such contracts does not currently exist and, infact, is unlikely to come into being, because market participants areused to locational prices for electricity, not congestion prices for thetransmission of electricity.

Other kinds of congestion compensation contracts include a FixedTransmission Right (FTR) available from PJM, which is a financialcontract that entitles the holder to a stream of revenues (or charges)based on a reservation level and hourly energy price differences acrossa specific path. The California Independent System Operator (ISO) has ahybrid contract called a “Firm Transmission Right” (also FTR) thatcombines features of FTRs and forward contracts. The markets for theseand other congestion compensation contracts are not as liquid as thefutures market and may be vulnerable to arbitrage. Moreover, spatialprice variation (i.e., basis risk) makes it difficult to evaluate theprice of congestion compensation contracts.

Therefore, there is a need for a technique to manage the price risk forelectricity at a particular location that both uses liquid price riskinstruments and accounts for spatial price variation. There is also aneed for a method of evaluating the price of congestion compensationcontracts and other price risk instruments, including forward andfutures contracts. There also exists a need for identifying arbitrageconditions of price risk instruments for electricity, either to avoidbeing arbitraged or to profit from arbitrage.

SUMMARY OF THE INVENTION

These and other needs are addressed by the present invention, in which acombination of multiple price risk instruments, e.g. futures contracts,congestion compensation contracts, etc., is selected in a particularproportion that reduces or even eliminates the Spatial Price Riskassociated with congestion. In contrast with other techniques, the pricerisk instruments need not be traded at the location in which the marketparticipant is interested; thus, the market participant is free tochoose among the most liquid of the available price risk instruments,such as futures contracts and possibly FTRs.

The present invention stems from the realization that the Spatial PriceRisk is almost completely associated with the congestion prices ofpotentially congestible lines at a prospective time T in the future.From these congestion prices, the physics of the power flows dictatesthe pattern of locational prices of electricity within the network atthe prospective time T, enabling any portfolio or combination of pricerisk instruments in the electricity market to be evaluated. Morespecifically, it is discovered that the cost f for a portfolio y ofprice risk instruments with respect to a market participant's underlyingposition z in the market at the prospective time T, can be described bythe following equation:f=(z′A−y′P′A)λ+y′F,  (1)

where A represents distribution factors describing the physics of thepower flows in the network as discussed in greater detail hereinafter, λrepresents the congestion prices of the congestible lines at theprospective time T, P represents the available market of priceinstruments (e.g. futures contracts, congestion compensation contracts,and other preferably liquid contracts in a market related toelectricity), and F represents the current prices (for delivery atprospective time T) of the price risk instruments.

Since the Spatial Price Risk is associated with the congestion prices λ,and since the cost F of the price risk instruments is currently known,the market participant's Spatial Price Risk can be reduced or eveneliminated by eliminating the role of λ in equation (1). In other words,to eliminate Spatial Price Risk the portfolio y of price riskinstruments should be chosen such that:z′A−y′P′A=0.  (2)

Accordingly, one aspect of the invention relates to a method andsoftware for managing risk in a market related to a commodity, such aselectricity, delivered over a network. Locational prices of thecommodity in the market are modeled as a linear combination ofcongestion prices for congestible lines in the network. Based on themodel, a combination of price risk instruments for the market isproduced in a proportion such that the effect of the congestion pricesfor the congestible lines on the locational prices of the commodity isreduced, or even eliminated.

Another aspect of the invention pertains to a method and software forevaluating a portfolio of price risk instruments in a market related toa commodity delivered over a network, such as electricity. A pluralityof distribution factors is estimated that indicates the effects on oneor more congestible lines in the network due to transfers of thecommodity at respective locations in the network. The portfolio is thenevaluated based on the estimated distribution factors. Other aspects ofthe invention involve a method for hedging a set of underlying positionsin the market and identifying arbitrage opportunities by producing aportfolio of price risk instruments for the market based on theestimated distribution factors.

Still another aspect of the invention is related to portfolios derivedby the above described methods or to any portfolio of price riskinstruments in which the price risk instruments are proportioned suchthat the effect of congestion prices of at least some congestible linesin the network on the prices of the commodity at locations in thenetwork is eliminated.

Other advantages of the present invention will become readily apparentfrom the following detailed description, simply by way of illustrationof the best mode contemplated of carrying out the invention. Theinvention is capable of other and different embodiments, and its severaldetails are capable of modifications in various obvious respects, allwithout departing from the invention. Accordingly, the drawing anddescription are illustrative in nature, not restrictive.

BRIEF DESCRIPTION OF THE DRAWINGS

The present invention is illustrated by way of example, and not by wayof limitation, in the figures of the accompanying drawings and in whichlike reference numerals refer to similar elements and in which:

FIG. 1 depicts an exemplary power network.

FIG. 2 is a flow diagram illustrating the operation of one embodiment ofthe present invention.

FIG. 3 is a flow diagram illustrating the operation of anotherembodiment of the present invention.

FIG. 4 depicts a computer system that can be used to implement anembodiment of the present invention.

DESCRIPTION OF THE PREFERRED EMBODIMENT

A methodology for managing risk in markets related to commoditiesdelivered over a network is described. In the following description, forthe purposes of explanation, numerous specific details are set forth inorder to provide a thorough understanding of the present invention. Itwill be apparent, however, to one skilled in the art that the presentinvention may be practiced without these specific details. In otherinstances, well-known structures and devices are shown in block diagramform in order to avoid unnecessarily obscuring the present invention.

In addition, the operation of embodiments of the present invention isillustrated with respect to an exemplary power network 100 shown inFIG. 1. The present invention is not limited to the exemplary powernetwork shown in FIG. 1, but is capable of application to other networkconfigurations and topologies, including existing power distributionnetworks comprising thousands of locations.

The exemplary power network 100 comprises, for purposes of explanation,five locations or “nodes”, 110, 120, 130, 140, and 150. Node 110 iscoupled to node 120 by transmission line 112 and to node 140 bytransmission line 114. Node 120 is further coupled to node 130 bytransmission line 123, to node 140 by transmission line 124, and to node150 by transmission line 125. Nodes 130 and 140 are further coupled tonode 150 by transmission lines 135 and 145, respectively.

For most examples discussed herein, it is further assumed that thereexists the possibility of transmission congestion in transmission line114 in the direction marked by the arrow, namely from node 110 to node140. Furthermore, it is assumed that two liquid forward electricitymarkets or futures markets exist, based on prices at the locations ofnodes 110 and 150, marked by a heavier line. Finally, it is also assumedthat there are no transmission losses. Departures from this basicexample, however, will be made to more fully explain the operation ofparticular embodiments of the present invention.

Modeling Locational Prices in the Network

FIGS. 2 and 3 are flow diagrams illustrating the operation of differentembodiments of the present invention, for example, to evaluate aportfolio of price risk instruments (see FIG. 2) or to generateportfolios of price risk instruments (see FIG. 3). In either case, thelocational prices in the network are modeled at an initial step 200 or300, respectively, based on the physics of power flows in relation tocongestible lines.

The present invention stems from the realization that the physics ofpower flows in an electric power network governs the pattern ofelectricity prices at the multiple locations in the electric powernetwork when one or more distribution and/or transmission lines arecongested, even at a prospective time T in the future. Morespecifically, the pattern of locational prices can be derived from alinear combination of the congestion prices and the price of electricityat one “reference” location. Thus, the locational spot prices in theelectric power network can be estimated by estimating the congestionprices, for example, by use of the following equation:S=Aλ,  (3)

where S represents the pattern of spot location prices in the powernetwork at the prospective time T, A represents distribution factorsdescribing the physics of the power flows in the network, and λrepresents the prices of congestion for the congestible lines at theprospective time T.

In one embodiment, A is an (l+1)×n sensitivity matrix, where lrepresents the number of congestible transmission lines, referred to as“flowgates,” and n represents the number of locations or nodes in thenetwork. The first column of A describes the effect of transmissionlosses on transfers of electricity. In particular, each entry of thefirst column of A equals one plus the percentage of transmission lossesthat occur when an incremental transfer of electricity is made between areference location and the location corresponding to the entry. In thespecial case when the power network has no transmission losses, thefirst column of A is all “ones.”

Each of the remaining columns of A contains Power Transfer DistributionFactors (PTDFs) corresponding to each of the flowgates. Each entry of aflowgate PTDF column is the percentage of the incremental flow in theflowgate that results from a transfer of electricity between thereference location and the location corresponding to the entry. Thefactors can be estimated from the relative impedance of each of thetransmissions, using a DC or AC load flow solution, and from sets ofPTDFs available from the North American Electric Reliability Council(www.nerc.com).

Referring to the exemplary network in FIG. 1, n=5 because there are fivenodes 110, 120, 130, 140, and 150, and l=1 because there is only oneflowgate, namely, congestible transmission line 114 in the directionfrom node 110 to node 150. Since it is assumed for purposes of examplethat transmission lines 112, 114, 123, 124, 125, 135, and 145 arelossless, the first column of A is all ones. The second column of Acontains the PTDFs corresponding to the flowgate 114. For example, if anadditional one megawatt is injected at node 110 and removed at node 120,and if 38% of the injected power will flow through transmission line114, then the corresponding entry for node 120 in the matrix A will be0.38. The examples discussed hereinafter assume for purposes ofillustration the following values in sensitivity matrix A:

$\begin{matrix}{A = {\begin{bmatrix}1.00 & 0.00 \\1.00 & 0.38 \\1.00 & 0.43 \\1.00 & 0.62 \\1.00 & 0.48\end{bmatrix}.}} & (4)\end{matrix}$

Furthermore, λ is an (l+1)-dimensional vector that characterizes thecongestion prices in the network, and may be based on the Lagrangemultipliers corresponding to the power flow equations and flowgatecongestion constraints. The first entry of λ is the spot price ofelectricity at the reference node, and the remaining entries of λ arethe prices of congestion with respect to the reference node of the lcongestible lines. When one or more of the l flowgates are congested,the corresponding entries in λ become positive, but if the flowgate isnot congested, the corresponding entry in λ is zero.

There is uncertainty in λ because there is uncertainty in whether therewill be transmission congestion in the future. On other hand, there islittle or no reasonable uncertainty in the sensitivity matrix A, becausemarket participants can infer the values of A from market observations,applications of the laws of physics, and the information published byNERC. Therefore, the uncertainty in locational prices is almostcompletely due to the uncertainty in λ, and the effect of spatial riskin one's portfolio can be reduced or even eliminated, by reducing therole of the parameter λ.

In the example, there is only one congestible transmission line,flowgate 114. Thus, λ would be a two element column vector [λ₁, λ₂]′,where λ₁ represents the spot price of electricity at node 110, and λ₂represents the congestion price for flowgate 114. If the spot price forelectricity at node 110 is λ₁=$20.00, and if there is no congestion inflowgate 114, i.e. λ₂=$0.00, then the spot prices in the network 100 isS=[20.00, 20.00, 20.00, 20.00, 20.00]′. This example shows that there isno spatial variation in price if there is no congestion.

On the other hand, if there is congestion on flowgate 114, then thesecond entry of λ assumes a positive value. Assuming that the spot pricefor electricity at node 110 is still λ₁=$20.00 but that the congestionprice for flowgate 114 is λ₂=$10.00, for example, the spot prices in thenetwork 100 become is S=[20.00, 23.80, 24.30, 26.20, 24.80]′. The higherspot price at node 140, S₄=$26.20, will encourage generators attached tonode 140 to supply more power at node 140, thereby reducing thecongestion on transmission line 114.

Evaluating Portfolios

Referring to FIG. 2, after modeling in step 200 the locational prices inthe network 100, a portfolio of price risk instruments is evaluatedbased on the model (step 202). As used herein, a portfolio is a set ofpositions to take financial advantage of particular market conditions orcharacteristics. The financial advantage may be to hedge against risk,in which the value of the portfolio is generally negative, or to assumerisk, in which the value of the portfolio hopefully is positive. Aposition is a specific asset or obligation traded in a market related toa commodity delivered over a network, such as the wholesale electricitymarket, the electricity derivatives market, and related markets. A pricerisk instrument refers to a position taken for delivery or settlement ata prospective time T in the future in the market and may include, forexample, forward contracts, futures contacts, congestion compensationscontracts, such as TCCs and FTRs, price swaps, basis swaps, optioncontracts, and other derivative contracts.

The m available price risk instruments for a network may be expressed asan n×m matrix P in terms of a weighted average of the prices at the nlocations in the market. A futures contract for delivering power at aparticular location would have a 1.0 for that location and a 0.0elsewhere. A TCC, on the other hand, would have a 1.0 at the locationwhere power is added and a −1.0 where power is removed. In the exampleof FIG. 1, there are two futures markets, one for delivering electricityat node 110 and another for delivering electricity at node 150. Thus, Pdefined for the example as:

$\begin{matrix}{P = {\begin{bmatrix}1.0 & 0.0 \\0.0 & 0.0 \\0.0 & 0.0 \\0.0 & 0.0 \\0.0 & 1.0\end{bmatrix}.}} & (5)\end{matrix}$

The market participant's underlying position in the market may be givenas an n-dimensional vector z indicating the buy and sell obligations ata prospective time T in the future. Positive values in z represent netload (or buy) obligations, and negative values represent net generation(or sell) obligations. Generally, z will have non-zero values for amarket participant who would like to hedge a particular underlyingposition, and all zeros for a speculator whose primary interest isassuming price risk.

The market participant's underlying position z at the prospective time Tin the future may be met by taking positions x in the spot market attime T, by taking positions y with price risk instruments (e.g. in afutures market now for delivery or settlement at time T), or acombination of both. Thus,z=x+Py.  (6)

The cost f to a market participant of a portfolio is given by (wherenegative f implies profits and prime denotes transposition):f=x′S+y′F,  (7)

where F is an m-dimensional vector that represents the prices of therespective price risk instruments. Combining equations (3), (6), and(7), the resulting formula for calculating the cost f of a portfolio yof price risk instruments in relation to the market participant'sunderlying position can be given by:f=(z′A−y′P′A)λ+y′F.  (8)

In the example, if a market participant has a set of buy/sellobligations z=[2, −1, 2, −3, 1]′, the cost for the market participant tomeet all the obligations in the spot market at time T (i.e. y=0) wouldbe f=z′Aλ. If the spot price for electricity at node 110 at time T is$20.00/MW and there is no congestion, then λ=[20.00, 0.00]′ and,therefore, f=$20.00. However, with the same spot price for electricityof node 110 of $20.00/MW, if there is congestion on transmission line114 with a congestion price of $10.00/MW, then λ=[20.00, 10.00]′ and,therefore, f=$11.00. The price swing of $9.00/MW shows that there isconsiderable Spatial Price Risk due to congestion.

If the market participant takes a long (buy) position of 1 MW at time 0for delivery at time T in the futures market for node 110, with theprices of futures given by F=[20.00, 22.00], then the cost in the firstscenario of λ=[20.00, 0.00]′ is also $20.00/MW, but the cost in thesecond scenario of λ=[20.00, 10.00]′ is $11.00/MW. Thus, there still isa price swing of $9.00/MW, showing that taking the position in a singleprice risk instrument has not reduced, in this example, the spatial riskdue to congestion.

Hedging

Referring to FIG. 3, another aspect of the present invention pertains tomodeling the electric power network (step 300) and then generating aportfolio to manage risk, for example, by hedging, constructing newprice risk instruments, and identifying low-risk arbitrageopportunities.

One embodiment of the invention therefore relates to hedging against therisk of an underlying position, which means reducing the price risk forfulfilling the underlying position at a prospective time T in thefuture. Referring back to equation (8), since the cost of the price riskinstrument F is known at time 0, the second term y′F is not subject torisk. All the spatial risk, thus, is associated with the uncertainty inthe congestion price vector λ. To eliminate the spatial risk, therefore,the portfolio y of price risk instruments is chosen such that term withthe congestion price vector becomes zero, orz′A−y′P′A=0.  (9)

To fulfill the remainder of the underlying position z at prospectivetime T, the market participant takes the position z−P y in the spotmarket at time T. The hedge costs y′F.

Equation (9) may be viewed as a linear equation in y, the marketparticipant's portfolio of price risk instruments. If m (the number ofprice risk instruments) is strictly less than l+1 (one more than thenumber of congestible transmission lines), then there are too manyequations in too few unknowns, and equation (9) will not usually besatisfied. In this case, however, the market participant may make a“partial hedge” by selecting a number of possible congestion events thatis less than the number of price risk instruments. Accordingly, themarket participant reduces exposure to the selected congestion events,while still remaining subject to the risk of the non-selected congestedevents.

On the other hand, if m is strictly greater than l+1, then there are toofew equations and too many unknowns, and equation (9) will be satisfiedfor many different portfolios y. In fact, the presence of too many mprice risk instruments will lead to the possibility of arbitrage.

Finally, if m equals l+1, there are as many equations as there areunknowns, and equation (9) will usually have a unique solution y.Equation (9) will also have a solution if z′=y′P′, which for anarbitrary z will have solution if and only if P is invertible or m=n.Therefore, complete hedging is available when the number of availableprice risk instruments is greater than the number of congestibletransmission lines, i.e. m≧l+1, or when the number of price riskinstructions equals the number of locations, i.e. m=n.

In the example, there are two futures markets (i.e., m=2) at nodes 110and 150, but only one congestible line 114 (i.e., l=1). Therefore, thereexists the possibility of complete hedging of the market participant'sunderlying position of buy/sell obligations z=[2, −1, 2, −3, 1]′.Solving for y in z′A−y′P′A=0, the solution y=[2.9, −1.9]′ is obtained. Avariety of matrix algebra and linear optimization techniques may beemployed to solve for y, for example, by use of the MATLAB™ softwarepackage, but the present invention is not limited to any particulartechnique.

Accordingly, the market participant would take a long (buy) position(2.9 MW) in the futures market at node 110 and a short (sell) position(−1.9 MW) in the futures market at node 150 for delivery or settlementat time T. The cost of the hedge is y′F=$16.20, which is the total costto the market participant if there is no congestion (e.g., λ=[20.00,0.00]′) or even if there is congestion (e.g., λ=[20.00, 10.00]′). Sincethe cost is always the same, notwithstanding the congestion of flowgate114, the Spatial Price Risk is completely eliminated. While the risk iscompletely eliminated, the hedge will not necessarily leave the marketparticipant better off than if the market participant had onlyparticipated in the spot market. In the example, with λ=[20.00, 10.00]′,the market cost is $11.00, but the hedge cost $16.20. The reason why themarket participant chooses to hedge, however, is for risk aversion, notprofit maximization.

Synthetic Price Risk Instruments

Hedging is generally favored by those market participants, such asutilities, that have a known underlying position in the actual commoditymarket in the form of power generation and load obligations againstwhich the market participant would like to eliminate price risk. Theprinciples described herein above, however, can also help brokers,dealers, financial institutions, and other financial services providersin offering new price risk instruments, even at locations that are notactively traded on the market.

A financial services provider who offers a new price risk instrument ineffect creates an underlying position z in the commodity and relatedmarkets reflective of the new price risk instrument. Thus, the financialservices provider can use equation (9) to hedge against that newunderlying position z with price risk instruments at other locations toeliminate the price risk for the financial services provider. The priceof the hedge will then determine the price at which the new price riskinstrument should be offered. The new price risk instruments may includeTCCs and other congestion compensation contracts at locations that arenot actively traded on the market. A buyer's choice contract, whichgrants the right to get power from any of a plurality of locations inthe network, can also be constructed.

In the example illustrated in FIG. 1, suppose a financial servicesprovider wishes to offer a 1 MW Transmission Congestion Contract (TCC)from node 130 to node 120. A TCC from one location to another pays thedifference in the price of electricity at the two locations, and can beused by others to hedge against risk. Thus, the underlying position ofthe TCC is z=[0, −1, 1, 0, 0]′. Using A and P defined in equations (4)and (5), respectively, and equation (9), the portfolio of existing pricerisk instruments to hedge against the TCC is chosen as y=[−0.1, 0.1]. Inother words, the financial services provider should take a long positionof 0.1 MW at node 150 and a short position of −0.1 MW at node 110. Thecost of this synthetic TCC would be y′F=$0.20.

Arbitrage

If a financial services provider is not careful in offering a new pricerisk instrument at the appropriate price, that financial servicesprovider may become vulnerable to arbitrage. Arbitrage is the purchaseof various financial instruments for a given time to profit from a pricediscrepancy. In accordance with one aspect of the present invention, anarbitrage condition exists among the electricity market related pricerisk instruments if there exists a portfolio y such that the followingconditions are satisfied:y′P′A=0 and y′F<0.  (10)

The first condition follows from equation (9), in which the arbitrageurhas no actual underlying position (i.e., z=0) in the market, but wishesto take a set of positions y to take financial advantage of the market.The second condition merely states that the cost of the arbitrageportfolio y is profitable. The first arbitrage condition is satisfiedwhen either the rows of P′A or the rows of P′ are linearly dependent,which generally occurs when m (the number of available price riskinstruments) exceeds l+1 (one plus the number of congestibletransmission lines) or exceeds n (the number of locations in thenetwork). In other words, if there are “too many” electricity price riskinstruments, then there is a possibility of arbitrage.

To illustrate, assume that, in addition to price risk instrumentsoffered in the example of FIG. 1, a 1 MW TCC is also offered from node140 to node 130 at a price of $0.10. Thus, the matrix of available pricerisk instruments becomes:

$\begin{matrix}{P = {\begin{bmatrix}1 & 0 & 0 \\0 & 0 & 0 \\0 & 0 & {- 1} \\0 & 0 & 1 \\0 & 1 & 0\end{bmatrix}.}} & (11)\end{matrix}$

Furthermore, assume that the future prices of these three markets areF=[20.00, 21.00, 0.10]′. The cost of the TCC is normally considerablylower than the energy supply contracts, because the TCC does not involveenergy per se, but rather the transfer of energy. If there is nocongestion, for example, the energy transfers offset at the same priceand the value of the TCC is zero. Based on these values of P, A, and F,it is possible to find a portfolio y of the three price instrumentsy=[1.0, −1.0, 2.5]′, such that closing out this position at time T willgenerate a guaranteed profit of $0.75. If, however, the TCC was offeredat the price of $0.40 (determined by the methodology described withrespect to creating synthetic price risk instruments), then the value ofthe arbitrage portfolio would be $0.00, indicating that the TCC isappropriately priced.

Unlike other market participants, such as utilities, arbitrageurs aregenerally willing to give up the elimination of Spatial Price Risk ifthey can always make a profit in every congestion scenario, even whenthe profit is variable. Accordingly, arbitrageurs would seek to find aportfolio y such that the following generalized conditions hold for allcongestion scenarios Λ:−y′P′Aλ+y′F≦0, for all λ in Λ  (12)

with the above inequality being strict for at least one member of Λ.

Options and Other Derivative Contracts

Options and other derivative contracts differ from the previouslydiscussed price risk instruments in that their value is not linear. Forexample, an option grants the right but not obligation to buy acommodity at a specified “strike” price. Thus, if the price of thecommodity is less than the strike price, then the option will not beexercised unless the option holder would like to take a guaranteed loss.On the other hand, the option holder will exercise the option if theprice of the commodity exceeds the strike price, because the optionholder can lock in a profit. Consequently, the valuation of an optionneeds to take into account the expected values of the commodity pricesto estimate the likelihood that the strike price will be exceeded.

Accordingly, an aspect of the present invention extends the evaluationof price risk instruments for options and other non-linear derivativesin equation (8) to account for the expected variation of congestionprices:f=(z′−y′P′)AÊ(λ(T))+y′F,  (13)

where Ê(λ(T)) represents a risk-neutral expectation for the congestionprices at time T. In one embodiment, the congestion is modeled by acombination of a Wiener process and a Poisson jump process to arrive atthe following analytical solution:

$\begin{matrix}{{{\hat{E}\left( {\lambda_{i}(T)} \right)} = {{{\lambda_{i}(0)}e^{{- \beta_{i}^{-}}T}} + {\left( {1 - e^{{- \beta_{i}^{-}}T} - {\frac{\beta_{i}^{-}}{\beta_{i}^{-} + \beta_{i}^{+}}\left( {1 - e^{{- {({\beta_{i}^{-} + \beta_{i}^{+}})}}T}} \right)}} \right){E\left( u_{i} \right)}}}},} & (14)\end{matrix}$

where β_(i) ⁻ and β_(i) ⁺ represent the mean arrival rates of thePoisson jump and death processes, respectively, which may be obtainedusing moment matching techniques on probability distributions of otheroptions, and E(u_(i)) represents an expected Poisson random jump size.

Hardware and Software Overview

In certain embodiments, execution of one or more steps in FIGS. 2 and 3may be automated on a computer system, which can be, for example, amainframe computer, minicomputer, workstation, personal computer, a webserver, a thin client, and an Internet appliance. FIG. 4 is a blockdiagram that illustrates a computer system 400 upon which an embodimentof the invention may be implemented. Computer system 400 includes a bus402 or other communication mechanism for communicating information, anda processor 404 coupled with bus 402 for processing information.Computer system 400 also includes a main memory 406, such as a randomaccess memory (RAM) or other dynamic storage device, coupled to bus 402for storing information and instructions to be executed by processor404. Main memory 406 also may be used for storing temporary variables orother intermediate information during execution of instructions byprocessor 404. Computer system 400 further includes a read only memory(ROM) 408 or other static storage device coupled to bus 402 for storingstatic information and instructions for processor 404. A storage device410, such as a magnetic disk or optical disk, is provided and coupled tobus 402 for storing information and instructions.

Computer system 400 may be coupled via bus 402 to a display 412, such asa cathode ray tube (CRT), for displaying information to a computer user.An input device 414, including alphanumeric and other keys, is coupledto bus 402 for communicating information and command selections toprocessor 404. Another type of user input device is cursor control 416,such as a mouse, a trackball, or cursor direction keys for communicatingdirection information and command selections to processor 404 and forcontrolling cursor movement on display 412. This input device typicallyhas two degrees of freedom in two axes, a first axis (e.g., x) and asecond axis (e.g., y), that allows the device to specify positions in aplane.

One embodiment of the invention is related to the use of computer system400 for trading. According to one embodiment of the invention, tradingis provided by computer system 400 in response to processor 404executing one or more sequences of one or more instructions contained inmain memory 406. Such instructions may be read into main memory 406 fromanother computer-readable medium, such as storage device 410. Executionof the sequences of instructions contained in main memory 406 causesprocessor 404 to perform the process steps described herein. One or moreprocessors in a multi-processing arrangement may also be employed toexecute the sequences of instructions contained in main memory 406. Inalternative embodiments, hard-wired circuitry may be used in place of orin combination with software instructions to implement the invention.Thus, embodiments of the invention are not limited to any specificcombination of hardware circuitry and software.

The term “computer-readable medium” as used herein refers to any mediumthat participates in providing instructions to processor 404 forexecution. Such a medium may take many forms, including but not limitedto, non-volatile media, volatile media, and transmission media.Non-volatile media include, for example, optical or magnetic disks, suchas storage device 410. Volatile media include dynamic memory, such asmain memory 406. Transmission media include coaxial cables, copper wireand fiber optics, including the wires that comprise bus 402.Transmission media can also take the form of acoustic or light waves,such as those generated during radio frequency (RF) and infrared (IR)data communications. Common forms of computer-readable media include,for example, a floppy disk, a flexible disk, hard disk, magnetic tape,any other magnetic medium, a CD-ROM, DVD, any other optical medium,punch cards, paper tape, any other physical medium with patterns ofholes, a RAM, a PROM, an EPROM, a FLASH-EPROM, any other memory chip orcartridge, a carrier wave as described hereinafter, or any other mediumfrom which a computer can read.

Various forms of computer readable media may be involved in carrying oneor more sequences of one or more instructions to processor 404 forexecution. For example, the instructions may initially be borne on amagnetic disk of a remote computer. The remote computer can load theinstructions into its dynamic memory and send the instructions over atelephone line using a modem. A modem local to computer system 400 canreceive the data on the telephone line and use an infrared transmitterto convert the data to an infrared signal. An infrared detector coupledto bus 402 can receive the data carried in the infrared signal and placethe data on bus 402. Bus 402 carries the data to main memory 406, fromwhich processor 404 retrieves and executes the instructions. Theinstructions received by main memory 406 may optionally be stored onstorage device 410 either before or after execution by processor 404.

Computer system 400 also includes a communication interface 418 coupledto bus 402. Communication interface 418 provides a two-way datacommunication coupling to a network link 420 that is connected to alocal network 422. For example, communication interface 418 may be anintegrated services digital network (ISDN) card or a modem to provide adata communication connection to a corresponding type of telephone line.As another example, communication interface 418 may be a local areanetwork (LAN) card to provide a data communication connection to acompatible LAN. Wireless links may also be implemented. In any suchimplementation, communication interface 418 sends and receiveselectrical, electromagnetic or optical signals that carry digital datastreams representing various types of information.

Network link 420 typically provides data communication through one ormore networks to other data devices. For example, network link 420 mayprovide a connection through local network 422 to a host computer 424 orto data equipment operated by an Internet Service Provider (ISP) 426.ISP 426 in turn provides data communication services through theworldwide packet data communication network, now commonly referred to asthe “Internet” 428. Local network 422 and Internet 428 both useelectrical, electromagnetic or optical signals that carry digital datastreams. The signals through the various networks and the signals onnetwork link 420 and through communication interface 418, which carrythe digital data to and from computer system 400, are exemplary forms ofcarrier waves transporting the information.

Computer system 400 can send messages and receive data, includingprogram code, through the network(s), network link 420, andcommunication interface 418. In the Internet example, a server 430 mighttransmit a requested code for an application program through Internet428, ISP 426, local network 422 and communication interface 418. Inaccordance with the invention, one such downloaded application providesfor trading as described herein. The received code may be executed byprocessor 404 as it is received, and/or stored in storage device 410, orother non-volatile storage for later execution. In this manner, computersystem 400 may obtain application code in the form of a carrier wave.

Accordingly, a system, method, software, and portfolios for managingrisk in markets relating to a commodity delivered over a network aredescribed. More specifically, techniques are disclosed, wherein a marketparticipant can construct portfolios of preferably liquid price riskinstruments in specific proportions such that the price risk for themarket participant's underlying position is reduced or even eliminatedfor contemplated congestion conditions. In addition to hedging, marketparticipants can construct and properly evaluate new price riskinstruments, as well as identify arbitrage opportunities among offerednew price risk instruments whose prices are not reflective of thephysics of power flows.

While this invention has been described in connection with what ispresently considered to be the most practical and preferred embodiments,the invention is not limited to the disclosed embodiments, but on thecontrary, is intended to cover various modifications and equivalentarrangements included within the spirit and scope of the appendedclaims.

1. A computerized method for managing risk in a market related toelectricity delivered over a network comprised of tradable networklocations, comprising the steps of: (1) a computer modeling locationalprices of electricity in the market as a linear combination ofcongestion prices for a plurality of congestible transmission lines inthe network, wherein said step of modeling locational prices comprises:determining a set of distribution factors representing the physics ofthe flow of electricity in the network, determining a plurality ofvalues representing the prices of congestion for the congestibletransmission lines at a prospective time, and determining a pattern ofspot locational prices in the network at the prospective time, whereinsaid pattern of spot locational prices is a function of said set ofdistribution factors and said plurality of values representing theprices of congestion for the congestible lines; (2) the computercreating a portfolio of future positions with respect to the set ofdistribution factors which includes: selecting a portfolio of price riskinstruments which represent the set of distribution factors describingthe physics of the flow of electricity in the network and the availablemarket for the price instruments; and (3) the computer producing acombination of price risk instruments with respect to the set ofdistribution factors for the market in which an underlying position inthe market is determined from: (a) the spot locational prices determinedin step (1), and (b) the portfolio of future positions with respect tothe set of distribution factors created in step (2), such that thedifference between the underlying position in the market with respect tothe set of distribution factors and the portfolio of future positionswith respect to the set of distribution factors is calculated such thatat least one amount of each of the price risk instruments areproportioned, thereby interlocking eventual locational prices andreducing an effect of the congestion prices for the plurality ofcongestible transmission lines on the locational prices of theelectricity.
 2. The method according to claim 1, wherein the step ofproducing the combination of price risk instruments further includes thesteps of: creating a table of congestion events with respect to thetradable network locations; populating the table with values for therelative impact on the locational price of each congestion event;creating from the table a portfolio of future positions with respect tothe set of distribution factors; assessing the risk of each of thefuture positions in the portfolio of future positions with respect tothe set of distribution factors by assessing the number of congestionevents that would result in a loss for the portfolio; and determiningfrom the assessment of risk which portfolio would result in the lowestrisk.
 3. The method according to claim 2, wherein the step of creating aportfolio of future positions with respect to the set of distributionfactors includes selecting a portfolio y of price risk instruments, suchthat:z′A−y′P′A=0, where A represents distribution factors describing thephysics of power flows in the network, P represents the available marketof price instruments, and z represents a market participant's underlyingposition in the market at the prospective time T, and wherein theportfolio includes a set of positions and primes denote transpositions.4. The method according to claim 1, wherein: the step of determining aset of distribution factors representing the physics of the flow ofelectricity in the network further comprises: determining a matrix A ofdistribution factors describing the physics of electricity flows in thenetwork, wherein said matrix A contains a column representing the effectof transmission losses on transfers of electricity and a column for eachof the congestible transmission lines representing the percentage of anincremental flow of electricity that results from the transfer ofelectricity between a reference location and each tradable networklocation; the step of determining a plurality of values representing theprices of congestion for the congestible transmission lines at aprospective time further comprises: determining a vector λ having valuesrepresenting the prices of congestion for the congestible transmissionlines at a prospective time T, wherein the vector λ contains one valueof the price of electricity at a reference location and a value for theprices of congestion with respect to the reference location and eachcongestible transmission line; and the step of determining a pattern ofspot locational prices in the network at the prospective time furthercomprises: determining a matrix S representing a pattern of spotlocational prices in the network at the prospective time T, whereinmatrix S is determined by the formula:S=Aλ.
 5. A computer-readable medium bearing instructions for managingrisk in a market related to electricity delivered over a network, saidinstructions being arranged to cause one or more processors uponexecution thereby to perform the steps of: (1) modeling locationalprices of the electricity in the market as a linear combination ofcongestion prices for congestible lines in the network, wherein saidstep of modeling locational prices comprises: determining a set ofdistribution factors representing the physics of the flow of electricityin the network, determining a plurality of values representing theprices of congestion for the congestible transmission lines at aprospective time, and determining a pattern of spot locational prices inthe network at the prospective time, wherein said pattern of spotlocational prices is a function of said set of distribution factors andsaid plurality of values representing the prices of congestion for thecongestible lines; (2) a computer creating a portfolio of futurepositions with respect to the set of distribution factors whichincludes: selecting a portfolio of price risk instruments whichrepresent distribution factors describing the physics of the flow ofelectricity in the network and the available market of priceinstruments; and (3) the computer producing a combination of price riskinstruments for the market in which an underlying position in the marketis determined from: (a) the spot locational prices determined in step(1), and (b) the portfolio of future positions with respect to the setof distribution factors created in step (2), such that the differencebetween the underlying position in the market with respect to thedistribution factors and the portfolio of future positions with respectto the set of distribution factors is calculated such that at least oneamount of each of the price risk instruments are proportioned, therebyinterlocking eventual locational prices and reducing an effect of thecongestion prices for the plurality of congestible transmission lines onthe locational prices of the electricity.
 6. A portfolio generatingsystem and portfolio comprising: a computer-based system configured togenerate a portfolio having a plurality of price risk instruments bycarrying out the following steps: (1) modeling locational prices ofelectricity in the market as a linear combination of congestion pricesfor a plurality of congestible transmission lines in the network,wherein said step of modeling locational prices comprises: determining aset of distribution factors representing the physics of the flow ofelectricity in the network, determining a plurality of valuesrepresenting the prices of congestion for the congestible transmissionlines at a prospective time, and determining a pattern of spotlocational prices in the network at the prospective time, wherein saidpattern of spot locational prices is a function of said set ofdistribution factors and said plurality of values representing theprices of congestion for the congestible lines; (2) creating a portfolioof future positions which includes: selecting a portfolio of price riskinstruments which represent distribution factors describing the physicsof the flow of electricity in the network and the available market ofprice instruments; and (3) producing a combination of price riskinstruments for the market in which an underlying position in the marketis determined from: (a) the spot locational prices determined in step(1), and (b) the portfolio of future positions with respect to the setof distribution factors created in step (2), such that the differencebetween the underlying position in the market with respect to the set ofdistribution factors and the portfolio of future positions with respectto the distribution factors is calculated such that at least one amountof each of the price risk instruments are proportioned, therebyinterlocking eventual locational prices and reducing an effect of thecongestion prices for the plurality of congestible transmission lines onthe locational prices of the electricity; the portfolio comprising: theplurality of price risk instruments for a market related to electricitydelivered over a network, wherein the price risk instruments y areproportioned such that z′A−y′P′A=0, A represents distribution factorsdescribing the physics of power flows in the network, P represents theavailable market of price instruments, z represents a marketparticipant's underlying position in the market at a prospective time T,and primes denote transpositions, wherein said computer-based systemcomprises: a communication mechanism for communicating information; aprocessor coupled to the communication mechanism for processinginformation; a dynamic storage device coupled to the communicationmechanism for storing information and instructions; and a static storagedevice coupled to the communication mechanism for storing staticinformation and instructions.
 7. The portfolio generating system ofclaim 6, wherein a number of the price risk instruments is greater thana number of the at least one congestible lines.